Ukrainian Mathematical Journal

, 63:580

# Properties of a certain product of submodules

• M. J. Nikmehr
• R. Nikandish
• S. Heidari
Article

Let R be a commutative ring with identity, let M be an R-module, and let K 1, . . . ,K n be submodules of M: We construct an algebraic object called the product of K 1, . . . ,K n : This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M n = M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M n is a projective (flat) R(M)-module.

## Keywords

Positive Integer Multiplication Module Prime Ideal Local Ring Maximal Ideal
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
M. M. Ali, “Idealization and theorems of D. D. Anderson,” Commun. Algebra, 34, 4479–4501 (2006).
2. 2.
D. D. Anderson and M. Winders, “Idealization of a module,” J. Commutative Algebra, 1, No. 1, 3–56 (2009).
3. 3.
D. D. Anderson, “Cancellation modules and related modules,” Lect. Notes Pure Appl. Math., 220, 13–25 (2001).Google Scholar
4. 4.
D. D. Anderson, “Some remarks on multiplication ideals,” Math. Jpn., 4, 463–469 (1980).Google Scholar
5. 5.
S. E. Atani, “Submodules of multiplication modules,” Taiwan. J. Math., 9, No. 3, 385–396 (2005).
6. 6.
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley (1969).Google Scholar
7. 7.
K. Divaani-Aazar and M. A. Esmkhani, “Associated prime submodules of finitely generated modules,” Commun. Algebra, 33, 4259–4266 (2005).
8. 8.
Z. A. El-Bast and P. F. Smith, “Multiplication modules,” Commun. Algebra, 16, 755–799 (1988).
9. 9.
C. Faith, Algebra I: Rings, Modules, and Categories, Springer, Berlin (1981).
10. 10.
J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York (1988).
11. 11.
H. Matsumara, Commutative Ring Theory, Cambridge University Press, Cambridge (1986).Google Scholar
12. 12.
M. Nagata, Local Rings, Interscience, New York (1962).
13. 13.
A. G. Naoum and A. S. Mijbas, “Weak cancellation modules,” Kyungpook Math. J., 37, 73–82 (1997).

## Authors and Affiliations

• M. J. Nikmehr
• 1
• R. Nikandish
• 1
• S. Heidari
• 1
1. 1.Toosi University of TechnologyTehranIran